How do find the only winning move for white in this position?

I would be interested to know how you approach such a position and your thought process behind it:

Well, first and foremost. This assumes there is a winning move without an engine analysis? How did you know there is one while playing? You probably don't.

Also there's a differennce between post analysis and in game analysis. Post analysis doesn't give you any time pressure.

So let's update the assumptions. How do you find the winning move without an engine and during an actual game with a limited time alotted?

So I would provide my answer as if I was playing rated with 2 mins on the clock. By the time I get to this positoon I already have a preconceived idea of what my opening and middle plans will be. So that will lead me in finding the right move HOPEFULLY. But I don't know your opening (I didn't check the whole game yet).

Second with the remaining time pressure, I will probably try to calculate some of the obvious moves. If I can't find any within 30 seconds, I will play a safe move. Anything that doesn't immedialy lose will be my right move.

I actually saw Be1, Qxf7, Ng6. There's a potential back rank mate or mate on the light squares but I can't calculate long enough to see the end. So with that in mind, I will pick a safe move.

Its fairly obvious the black queen is tied to the rook, so thinking about distraction/diversion whatever is natural. The eye is immediately drawn to the knight sac but it gives up a mate in 1. There aren't a lot of other ways to even think about bringing more pieces in. So if you have plenty of time, and your checking everything because you smell a tactic with the rook tied down, then maybe. But be1 is just such a sophisticated move, and as @noobie519 said it doesn't look obviously winning within the typical search depth of a non-master. And there are clearly safer moves that still give a big advantage.

19. Be1 is very hard to calculate correctly, especially at this time control (basically you need to see the fork with Qd8+ and Qd4+ at the end of the Nxd1xe3xc4 line to be sure it's great). Move by move alternatives like 19. Rf1 Ng4 20. Rh1 h6 21. Be1 and 20. Nd1 intending Bc3 are natural and strong, so finding it is not compulsory.

As a puzzle, or as a live game? In a live game, I don't bother; I play Rf1 and then maybe Rh1. The rook is pinned, the queen is tied up defending it, and White is swinging ever more pieces to the kingside attack. I might see Be1 after Rf1 or Rh1, just as another way to get a piece over kingside.

As a puzzle...well I probably get it wrong, but the obvious target is that rook, and the only piece that can defend it is the queen. So you look for the possible spaces the queen can defend it from, and you see it's only f6 and e7; anywhere else and the rook falls. Once you see that, you might, eventually, realize that Bh4 would take away all of the queen's defensive squares, while something like Bc3 only takes away one and leaves the other. At that point, you can play Be1 without caring about the rook, because Black doesn't actually have the time to capture it without Bh4 wrecking them.

Be1 with threat of Bh4 wins. I thought for 1-2 minute